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Puzzle-solving in Psychology: the Neo-Galtonian vs.Nomothetic Research Focuses

Stéphane Vautier, Emilie Lacot, Michiel Veldhuis

To cite this version:Stéphane Vautier, Emilie Lacot, Michiel Veldhuis. Puzzle-solving in Psychology: the Neo-Galtonianvs. Nomothetic Research Focuses. New Ideas in Psychology, Elsevier, 2014, 33, pp.46-53.<10.1016/j.newideapsych.2013.10.002>. <halshs-00877010>

Puzzle-solving in Psychology: the Neo-Galtonian vs.

Nomothetic Research Focuses

Stephane Vautiera, Emilie Lacota, Michiel Veldhuisb

aUniversity of ToulousebUtrecht University

Abstract

We compare the neo-Galtonian and nomothetic approaches of psychologicalresearch. While the former focuses on summarized statistics that depict av-erage subjects, the latter focuses on general facts of form ‘if conditions thenrestricted outcomes’. The nomothetic approach does not require quantifica-tion as a convenient way of statistical modeling. The nice feature of a generalfact is its falsifiability by the observation of a single case. Hence, as a clearsense of scientific error is re-introduced in the research paradigm, we detailtwo kinds of puzzle-solving: repairing general facts by contraction or by ex-pansion of the initial conditions. This style of research does not require thatresearchers depend on highly skilled engineers in data analysis, as the verystructure of a general fact can be established by scrutinizing a contingencytable.

Keywords: methodology, measurement error, general fact, falsifiability

The present article contrasts two research focuses in Psychology. As theseresearch focuses define two classes of goals, we will call them paradigms. Thefirst one has been called neo-Galtonian by Danziger (1987, 1990) and is mostgenerally used in mainstream methodology in psychology, while the secondone can be properly called nomothetic, as argued by Lamiell (1998) andVautier (2011, 2013), and constitutes a blind spot of psychological research.Let P1 and P2 denote neo-Galtonian and nomothetic research, respectively.To describe concisely these two paradigms, the mathematical concept of arelation will be useful.1 A (binary) relation from a set A on a set B is defined

1According to French philosopher of science Gilles-Gaston Granger (1995), “Scientific

Preprint submitted to New Ideas in Psychology October 25, 2013

Neo‐Galtonian (P1) prediction is based on a

false functionfalse functionf(X) = Ŷ

X’s permissible values Y’s permissible valuesNomothetic (P2) predictionis based on a general factg

X Y

The independent variable Xis a descriptive function

The dependent variable Y isa descriptive function

Statisticalpopulation

Figure 1: The common statistical structure of neo-Galtonian and nomothetic forms ofprediction, and their specific features.

by a set of ordered pairs (a, b) such that a is in A (a ∈ A) and b is in B.(This set is called the graph of the relation.)

Whereas the P1’s researcher is used to being satisfied with relations thatemerge from aggregates of persons but are logically irrelevant to depict phe-nomena at the scale of single persons (see, e.g., Danziger, 1987; Krause, 2011;Lamiell, 2003, 2013; Molenaar, 2004), the P2’s researcher inspects the samedata to discover a special case of relations, what Vautier (2013) calls generalfacts, which are, by definition, true for any person. As depicted in Figure1, an easy to spot difference would be that, whereas P1 focuses on expectedpoint-values, P2 focuses on a necessary set of values. P1’s slogan may beexpressed as follows:

Y = f(X) + E, (1)

knowledge based on experience always consists of constructing schemas or abstract modelsof this same experience and uses the relations between the abstract elements of these mod-els to deduce–through logic and mathematics–properties that correspond with sufficientprecision to directly observable empirical properties.” (p. 95, authors’ translation)

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where Y is the dependent variable, X is the independent variable and de-scribes initial conditions, f is a function of these conditions, and E denotesa random component that obeys a convenient probability law. The crucialconcept here is that of a function, that is, the special case of a relation suchthat f(X) is a single value2.

Contrastingly, P2’s slogan can be formulated as follows:

X ∈ α⇒ Y ∈ β, (2)

where α and β are non-empty strict subsets of X and Y ’s sets of permissiblevalues (the respective codomains of X and Y ). Thus, although P2 does notpreclude a functional relationship, it does not expect it. What is especiallyexpected is that the descriptive device provided by X and Y ’s codomainswill suggest that there is a least one subset of initial conditions which worksas a sufficient condition for a strict subset of Y ’s codomain.

What is at stake is to move freely from one research paradigm to an-other, instead of defining opposite and mutually ignoring camps. Within P1,psychologists are used to manipulate propositions pertaining to ‘constructs’(Cronbach & Meehl, 1955); for critical views, see Maraun (1998) and Michell(2013). As a corollary, P1-psychologists are trained to accept the institu-tional division of labour between the psychologist, whose expertise pertainsto substantive theory, that is, the realm of constructs or the ‘nomologicalnetwork’, and the statistician (or psychometrician), whose expertise dealswith data analysis. A catastrophic consequence of this division of labor isthat the sense of scientific error within the discipline becomes more and moreesoteric or even socially irrelevant (see, e.g., Borsboom, 2006). This is why,if psychologists are motivated to make a science of their discipline (see, e.g.,Borsboom et al., 2009; Lilienfeld, 2010; Vautier, 2011), they have to takeintellectual responsibility for its epistemology and methodology. Within P2,division of labor is superfluous. However, the required intellectual style obeysMonsieur Teste’s injunction: “Always demand proof, proof is the elementarycourtesy that is anyone’s due” (Valery, 1973, p. 65).

In the first section of the present article, it is argued that P1’s sloganmimics functional prediction in the natural sciences, but the price to be paid

2A (statistical) variable is also a relation, which is defined from the set of a population–called its domain–on the set of its admissible values–its codomain. The population is thedomain of X and Y . The function f is a relation from the codomain of the independentvariable X on the codomain of the dependent variable Y .

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is that the sense of approximation that characterizes the natural sciencesis lost. P2’s slogan rejects the functional imperative by acknowledging ir-reducible indetermination of prediction and instead hypothesizes restrictedapproximation. The second section opposes the infalsifiability of predictivestatements in P1 to the falsifiability of predictive statements in P2. The thirdsection exposes two kinds of puzzle-solving P2-researchers have to deal with.

1. Prediction: Restricted vs. Unrestricted Range of Approxima-tion

Let us take Kuhn’s (1996) words to get a sense of what empirical approx-imation means in this context:

Perhaps the most striking feature of the normal research prob-lems we have just encountered [in the physical sciences] is howlittle they aim to produce major novelties, conceptual or phenom-enal. Sometimes, as in a wave-length measurement, everythingbut the most esoteric detail of the result is known in advance,and the typical latitude of expectation is only somewhat wider.Coulomb’s measurements need not, perhaps, have fitted an in-verse square law; the men who worked on heating by compressionwere often prepared for any one of several results. Yet even incases like these the range of anticipated, and thus of assimilable,results is always small compared with the range the imaginationcan conceive. And the project whose outcome does not fall inthat narrower range is usually just a research failure, one whichreflects not on nature but on the scientist. (p. 35)

In other terms, the paradigmatic slogan of normal quantitative science ex-presses as

y = f(x) + ε, (3)

where y and x are quantitative variables, f is a numerical function, and εdenotes an approximation component within a restricted 100% confidenceinterval. By ‘restricted’, we mean that the range of the expected outcomesenables the researcher to eliminate a wide subset of logically possible out-comes that the measurement procedure allows.

For example, we can determinate that Paul’s height today lies between1.74 and 1.76 cm, and this range of approximation is compatible with the

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claim that the probability to find that, in suitable measurement conditions,Paul’s height today is not in this interval is null. If it was the case, such afinding would merely reflect a measurement aberration–not a measurementerror. In P1-psychology, such a finding would be qualified as measurementerror. To make the point clear, it suffices to remark that the random com-ponent in Equation (1) takes its values in the set of the possible values ofthe dependent variable. Usually, this variable is given a Gaussian probabilitydensity function, which is valued in ] −∞, +∞[. Hence, this approach toprediction precludes aberrant outcomes: any outcome is possible.

It is a striking paradox that quantitative psychology, while having adaptedits paradigmatic slogan from Equation (3) to Equation (1), has concededthe epistemological principle of unpredictability of its dependent variables.Consequently, the concept of (restricted) approximation used in the naturalsciences has been lost. The prediction Y = f(X) becomes a better expec-tation, yet preserving practical utility for decision making in mass settings.Imitation of quantitative laws saves the form of approximative prediction inEquation (3), as Y is a real value, but ratifies total ignorance at the scaleof the observation unit since the approximation of prediction is now unre-stricted.

The paradigm P2 goes one step further, as its typical goal consists inre-introducing the sense of restricted approximation in psychological predic-tion. There is a double issue at stake. First, if it is acknowledged that P1-psychology is valid only for mass psychology, there is a need for a genuinelygeneral psychological science, i.e., a science valid (and hopefully, sound) forany psychological system. In clinical psychology, in cognitive psychology, indevelopmental psychology, psychological systems correspond to single per-sons and not to collectives of persons. Even in social psychology, the unit ofanalysis is a person whose behavior is analyzed with respect to a group of per-sons. However, P1-psychology admits that the single person is unpredictable.For example, according to Rasch (1960),

Where it is a question of human beings and their actions, it ap-pears quite hopeless to construct models which can be useful forpurposes of prediction in separate cases. On the contrary, whata human being actually does seems quite haphazard, none lessthan radioactive emission. (p. 11)

In the perspective of P2-psychology, such a claim deserves verification. Thus,the target of P2-psychology consists in predictions with restricted approx-

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imation. Importantly, usefulness of P2’s program for social engineering inmass settings should not be mandatory, as restricted approximation meansexactly that what is to be predicted is a strict subset of values (which doesnot entail that its cardinality is one–in which case the prediction would be apoint value). Thus, a priori, psychological phenomena that can be describedby psychologists do not warrant the degree to which approximation can bereduced and the fact that approximation is restricted is a starting hypothesis.

Coming back to Equation (1), the aim is to discover initial conditions(pertaining to X) that warrant bounded residual variability, in such a waythat knowing the initial conditions, it would be possible to exclude at leastone admissible value of the dependent variable (Y ). Such aim involves a re-versal of perspective: whereas Equation (1) focuses on the positive, punctual,necessary, and in fact false value Y given X, Equation (2) focuses on a strictnecessary and hopefully true subset β given X. P2’s perspective reconcilespsychological research and what constitues for us the very wisdom of scien-tific knowledge, which consists in admitting and recognizing various formsof objective impossibility. Indeed, Equation (2) focuses equivalently on thecomplement set β of empirical impossible values given X. Let M(Y ) denoteY ’s codomain (set of admissible values). The existence of β implies that ofits non-empty complement β:

M(Y ) \ β = β. (4)

Thus, P2 focuses on a set β of empirically impossible, although logically pos-sible, events conditionally to given initial conditions identified as α, a strictsubset of X’s codomain. The reversal of focus from necessity to impossibilityechoes Popper’s (1959) profound remark: “[natural laws] do not assert thatsomething exists or is the case; they deny it” (p. 48).

In addition, there is no need to postulate that natural laws are quantita-tive, and the dependent variable Y may be qualitative in “nature”–althoughthe definition of the codomain of any function Y is not natural but cultural(Danziger, 1990).3 This is the second issue at stake with adopting P2’s re-search focus. To sum up, the shift from P1 to P2 is double: quantitative

3“But in truth scientific psychology does not deal in natural objects. It deals in testscores, rating scales, response distributions, serial lists, and innumerable other items thatthe investigator does not find but constructs with great care. Whatever guesses are madeabout the natural world are totally constrained by this world of artifact” (p. 2).

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dependent variables are replaced by qualitative ones, and the expected pre-diction, a point-value, is replaced by a necessary strict subset of values.

2. Testability of Predictions

As far as Equation (1) comprises the random component E, and giventhat E’s distribution is usually thought of as Gaussian, it is clear that a state-ment of form (1) denies nothing: given any image of X (or any observed valueon X), Y may take any value in its codomain M(Y ). Actually, the normaldistribution of E is a crude and logically unsuitable approximation becauseM(Y ) is bounded and discrete (e.g., a test scale is a finite numerical series,see Vautier et al., 2011, 2012). The crucial point is that the random vari-able is defined in such a way that its range of possible values is unrestricted,that is, it implies no impossible values in M(Y ). As “it is a fallacy to inferan individual propensity from a statistical analysis of the distribution of aproperty in a population” (Harre, 2004, p. 7), the random variable E playsthe role of an interpretative concept, by contrast to a falsifiable probabilistichypothesis about residual variability associated with the observable responseof a class of observation units from the population. Thus, P1 focuses on theappearance of punctual predictability (for the sake of social utility) and losesfalsifiability at the level of a class of observation units induced by a givenvalue on X.

The paradigm P2 focuses on falsifiability, even if it means that punctualpredictability may be lost as the primary motivation of the research project–which does not entail that punctual estimation is lost if this kind of goal isto be pursued for other non-scientific reasons. The logical form of Equation(2) implies that if it is true, the following statement is true as well:

Y ∈ β ⇒ X ∈ α. (5)

As X and Y ’s values are observables, Equation (2) is falsifiable: it is logicallypossible to observe

Y ∈ β & X ∈ α, (6)

in which case Equation (5) is false and hence Equation (2) is also false.

3. Puzzle-solving

Kuhn (1996) describes normal research as puzzle-solving. We have diffi-culties to figure out how normal research in P1 could be described as puzzle-solving at the scale of classes of single cases defined by their common value

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on X, as unrestricted approximation is normal. Instead, we understandnormal research in P1 as a kind of fuzzy process of “translation” of non-empirical ideas–that hardly match the concept of falsifiable theory–into sta-tistical statements that depict the average subject. A characteristic feature ofP1 is that (substantive) theory is intrinsically disconnected with description(and hence genuine measurement) of real persons. Theoretical explanationconsists mainly in elaboration of abstract representations about inobservablesand demonstration of confirmatory operationalized evidence at the level ofaggregates (Danziger, 1987; Vautier, 2011). In P2, theoretical explanation(or invention, Hempel, 1966) has no object until one empirical law has beenformulated. We do not know of a single report in psychological literature har-boring an empirical statement of form (2). Actually, nomological networksof psychological constructs advocated by Cronbach & Meehl (1955) workas pre-theoretical or pre-scientific concepts. This is because scientific con-cepts require empirical regularities to have an empirical meaning. The taskof nomothetic (P2-)psychology consists of discovering descriptive approachesable to reveal lawful–and non trivial–phenomena. In the meantime, normalscience in nomothetic psychology consists of indexing candidates for lawfulstatements, and then in indexing and elucidating their associated counter-examples.

To analyse the puzzle-solving’s structure in P2, it will be useful to detailhow the variables X and Y are functions (cf. Figure 1). Their commondomain is a discrete set Ω = ωi, i = 1, . . . , and their codomains areM(X) and M(Y ), respectively:

X : Ω −→ M(X)Y : Ω −→ M(Y )

(7)

The important concept is that of the empirical support Ω. The population isa set of observation units of form ω = (u, t), which denote a person u locatedin time t. There are two possible starting points. The first one is a generalfact (Vautier, 2011, 2013), that is, the triplet (Ωn, α, β) such that

∃ α ⊂M(X), β ⊂M(Y ),∀ ω ∈ Ωn ⊂ Ω, X(ω) ∈ α⇒ Y (ω) ∈ β. (8)

In other terms, the triplet (Ωn, α, β) represents a non tautological fact, ofwhich the generality is bounded to the n objects (or observation units) of theworld Ωn. The general fact stipulates sufficient conditions, namely havingthe property α, for having the property β.

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Table 1: Conjoint distribution of test scores and illness state.

Test scores i ¬i[40, 71] 26 2[72, 87] 35 51[88, 94] 0 40Note. i = illness, ¬i = non illness.

Let us take the example of a memory test the total score from which isused to predict the presence of illness i (Lacot et al., 2011). Thus, M(X) isa univariate set of test scores; specifically, M(X) = 40, 41, . . . , 94. AndM(Y ) = i, ¬i. The data here update Lacot et al.’s data with a sample of154 participants. Table 1 exhibits the general fact

∀ ω ∈ Ω154, X(ω) ∈ [88, 94]⇒ Y (ω) = ¬i, (9)

which suggests a law on the parent population Ω.Now, a counter-example is the new observation unit ω155 ∈ Ω \Ω154 such

thatX(ω155) ∈ α & Y (ω155) ∈ β, (10)

where α = [88, 94] and β = i. Thus, the counter-example ω155 does notfalsify the general fact (Ω154, α, β) but the law (Ω, α, β).

Repairing the law is the P2-puzzle. Such problem has the same structureas that of repairing what we call a pseudo-general fact; which constitutes thesecond possible starting point. In research practice, the latter may be morerelevant than the former. A pseudo-general fact is a general fact sufferingfew counter-examples. Table 1 exhibits the following pseudo-general fact:

∀ ω ∈ Ω154, X(ω) ∈ [40, 71]⇒ Y (ω) = i, (11)

which is false because of two counter-examples. Let us note them ω153 andω154.

Thus, the puzzle is to repair the general fact (11). Levi (1980) suggeststwo repairing approaches. The first one is contraction of M(X) (see Figure2, upper panel), while the second one is its expansion (see Figure 2, lowerpanel).

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M(X)

αContraction

α’

M(X)( )

α

M(X’’)M(X)

α’’α

Expansion

Figure 2: Two strategies for law repairing: Contraction and expansion of the validitydomain of a law (in gray, specification of the initial conditions). α = the initial conditionsfor a law. α′ = the contracted conditions (where the initial conditions have been adapted–contracted–to exclude falsifying cases). α′′ = the expanded conditions (where the initialconditions have been expanded, with relevant descriptive variables, to exclude falsifyingcases). M(X) = the codomain of X (with the initial conditions). M ′′(X) = the codomainof X ′′ (with the expanded conditions).

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3.1. Contraction

Contraction aims at restricting the initial conditions to save their suffi-cient character. Let Ωn′ be the set of counter-examples, and M ′(X) the setof their images by X. Coming back to our example, we have to consider thatΩn′ is comprised of the elements ω153 and ω154. The data reveal that thesetwo exceptions had test scores of 51 and 67 (M ′(X) = 51, 67). The ideaconsists of removing these specific initial conditions from the statement (11).Let α′ = α \M ′(X). Thus,

∀ ω ∈ Ω154, X(ω) ∈ α′ ⇒ Y (ω) ∈ β. (12)

Repairing (Ω154, α, β) yields (Ω154, α′, β), where α′ ⊂ α (and, trivially,

α′ 6= ∅).Coming back to the example, we get α′ = [40, 50] ∪ [52, 66] ∪ [68, 71].

However, the substantive meaning of such repairing has to be assessed. Weget no insight in removing the specific values 51 and 67 from the initialconditions because we are not ready here to question the assumption thatthe test scores measure an ordinal attribute. If we expect that the dependentvariable depends on certain levels of the attribute in such a way that lowlevels predict illness, it does not make sense to admit that the level 70 forexample predicts illness whereas the level 67 does not predict illness, becausethe level 67 is also a low level. Consequently, the contraction approach maybe useful only if it makes sense to restrict the initial conditions. Anotherway of contracting the initial conditions would be to retain only the subsetof levels [40, 50] and to admit that the intermediate subset which providesno prediction is redefined as [51, 87]; this would preserve the intuition thatthere are low levels of X that predict illness.

Some readers (like one of the anonymous reviewers of the present paper)could ask for the difference between “mere theory disconfirmation” and “theprogrammatics of fashioning of better theory”. Let us suppose that thesetwo cases are out of doubt (if they were, they should not be considered asfalsifying cases). Do the two discrepant cases falsify any theory? They fal-sify the general claim stated in Equation (11). Do we have any theory thatimplies this general fact? We have only the intuition that low levels shouldpredict illness, if no other conditions that are ignored in the definition ofX’s codomain are relevant, and we also admit that other conditions may berelevant, at least at certain levels of X. Thus, we should not be surprised bythe finding of falsifying cases because we know that the descriptive strategy

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which consists in considering that all relevant initial conditions are taken intoaccount in X’s codomain is naive. Moreover, test scores are not undisputabledescriptive values: In a nutshell, the test scores are the names of equivalenceclasses of multivariate descriptions (usually, the responses to the test items),and the ordering that is imposed on these classes via their numeration (scor-ing) is a disputable convention, a fact that is not new (Johnson, 1935, 1943)but remains usually ignored (Michell, 1990, 1997, 2000). We cannot discussthe theoretical and methodological consequences of such criticism in this pa-per (but we do it elsewhere). Thus, if we maintain the intuition that, ceterisparibus, low levels on X should imply illness, and because the ceteris paribuscondition is not controlled, contraction may be useful for prediction basedboth on evidence and intuition if we assume that within the contracted range,which is [40, 50], some ‘factors’ that are relevant out of this range can beneglected within this range. Of course, such reasoning is perfectly inductive(ad hoc) and thus requires further testing. If further tests were to corrobo-rate the contracted statement, there is no need to propose potential relevantfactors within this range and then for raising the issue of their empirical de-scription; if these tests falsify it, contraction failed and the issue consists ofdiscovering the relevant factors, which yields expansion.4

3.2. Expansion

From the point of view of the known initial conditions, any observationunit ω is described in M(X). For the sake of simplicity, we have chosenour working example because X offers a univariate description. To illus-trate the logic of expansion, we will consider only additional descriptions(potential relevant factors) that were ignored in X’s codomain definition,but are nevertheless available in the data (viz., gender, age, and educationallevel). Thus, considering additional descriptive factors (or dimensions ac-cording to Krause, 2010) in the initial conditions yields to considering amultivariate independent variable, that is, a vector of descriptive functionsX1X2 . . . Xm, in which case its codomain is the m-ary Cartesian productM(X1)×M(X2)×· · ·×M(Xm), where m is the number of factors that serveto describe the observation units’ initial conditions.

The idea of expansion as opposed to contraction consists of discoveringan independent variable, to be denoted by X ′′, that is useful to specify the

4An other consequence is that the theoretical and yet descriptive grounds of the testscores could be seriously questioned, but we ignore this way here.

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counter-examples with respect to the initial conditions. The dimensionalityof the codomain M(X ′′) is unknown and ranges in [1, 2, . . . , m − 1]. Letus supposes that X ′′ does exists and let α′′ be the subset that contains theimages of the counter-examples in M(X ′′). The updated initial conditionsare

α′ = α ∪(M(X ′′) \ α′′), (13)

and the independent variable becomes XX ′′. Repairing (Ωn, α, β) yields(Ωn, α

′, β).Coming back to the example, the two counter-examples share specifically

the conjonctive property of being female, more than 83 years old, and nothaving studied at the secondary educational level. The word “specifically”is crucial here because it is necessary that the remaining observation unitsdo not possess this property. Let α′′ denote this descriptive state–X ′′ is a3-variate function based on gender, age, and educational level. Thus, thepseudo-general fact (11) may be repaired as the following general fact:

∀ ω ∈ Ω154,(X(ω) ∈ [40, 71] & X ′′(ω) /∈ α′′)⇒ Y (ω) = i, (14)

The preceding statement highlights the fact that expansion consists insaving a general statement by isolating its falsifying cases, which consistsof adding negative requirements (e.g., not being female and more than 83years old and having a ‘low’ educational level) to the initial conditions (e.g.,having a low test score or, in other terms, the test score is in [40, 71]). Likecontraction, expansion is an inductive solution to the falsifying cases. Doesit help improve theory? Not at all because there is no theory to predict thegeneral statement to be saved. Hence, there is no theory to save or improveat all. In our understanding of what a scientific theory is, it is an abstractrestatement of natural or experimental, or empirical laws that articulate theselaws with parsimony (see Duhem, 1991). Consequently, there is no scientifictheory in a field of research if this field contains no law–just intuitions. This iswhy the main stake of scientific research in psychology consists in discoveringgenuine laws, and hence, to discover general facts. This is why we believethat the formalization of a general fact deserves to be known by nomotheticresearchers, as it is more difficult to look for a general fact if one ignores itsdefining property (Vautier, 2011). The ability to describe a data set by astatement of form (2) is a good test for deciding whether the data reveals ageneral fact.

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3.3. Known cardinality of a general fact

The known cardinality of an empirical general fact can be simply mea-sured by the number of its relevant observation units: Each time a newobservation unit falling into the class α falls into the class β (i.e., a posi-tive observation), or each time a new observation unit falling into the classβ falls into the class α (i.e., a negative observation), the known cardinalityincrements of one unit. It could be useful to introduce the following notation:

(Ω, α, β)[c1, c2], (15)

where c1 and c2 denote the number of ‘positive’ and ‘negative’ observations,respectively, and c1 + c2 measures the known cardinality.

A general fact repaired by contraction of α to α′ inherits the cardinalityof the equivalence class induced by α′ on Ωn. When a general fact has beenrepaired by expansion, two situations have to be distinguished. First, ifit can be verified that the remaining observation units do not possess theisolating properties α′′, the cardinality of the repaired fact is that of itspredecessor minus the number of counter-examples. But as the descriptivevariable used to isolate the counter-examples may be new, which yields thesecond situation, the descriptive status of the remaining observation unitswith respect to X ′′ is unknown. Consequently, the researcher has to supposethat the remaining observation units do not possess the isolating propertyα′′. As this supposition is uncertain, the cardinality of the repaired generalfact is reset, hence the triplet (Ω, α′, β)[0, 0].

4. Discussion

In the neo-Galtonian paradigm, researchers investigate summarized prop-erties of collectives of persons, which describe the average subject, that is,nobody in particular. Consequently, individual observations cannot play therole of falsifying cases in this paradigm; this is the reason why statistical P1

models hardly match the notion of an empirical law, but work much as inter-pretative devices interfacing “hard data” and “soft ideas” via conventionaloperationalization (Vautier, 2011).

The search for empirical laws requires a descriptive language of formM(X)×M(Y ) and a population Ω, from which observations can be made–an observation is a basic statement of form XY (ω) = xiyj. If there is a law

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in [Ω, M(X)×M(Y )], it has the form (Ω, α, β), where

1 ≤ |α| ≤ |M(X)|,1 ≤ |β| < |M(Y )|.

(16)

A law is a special relation that can be characterized by the fact that its graphG is a strict subset of M(X)×M(Y ):5

G ⊂M(X)×M(Y ). (17)

Let us call |β| − 1 the degrees of freedom (dfs) of the law (Ω, α, β). Bydefinition, if df = 0, the law is totally deterministic; otherwise, it is partiallydeterministic. As an anonymous reviewer of a previous version of this articlesuggested, one may be attracted by reducing the dfs of the law. However, wewill argue that this line of ‘puzzle-solving’ is programmatically premature, asit supposes that a law of positive dfs has already been established. It seemsnot an exaggeration to state that such laws are rare in psychology. Even ifone proposes a law with too many dfs (df > 1), the first task is to test it. Ifit is easy to find falsifying observations, there are no dfs to reduce becausethere is no law. Then, the logic of testing suggests the P2-researcher to focuson the search for falsifying cases, or at least for a practically sufficient amountof corroborating evidence if no falsifying cases can be found.

When the starting point of a corroborated law is available, the reducingtask to improve the granularity (or precision) of prediction will consist inrefining the description of the initial conditions, in order to find ‘nomotheticmoderators’.6 However, as long as psychological knowledge is framed in theP1-language of radical indeterminacy–there is no restriction on the permis-sible values of any phenomenon given any initial conditions–, such startingpoints will be lacking. If the proposed law is not trivial, the law is unlikely,and it is likely that it will be falsified, which entails that the main scientifictask will be to repair it rather than to restrict its dfs.

5The graph of the relation from M(X) to M(Y ) as it is known with respect to theavailable evidence from the sample Ωn is defined as follows:

G = (xi, yj)| ∃ ωk ∈ Ωn, X(ωk) = xi & Y (ωk) = yj,

where i and j index X and Y ’s observed values, respectively.6Krause’s (2010) approach to the search for sufficient condition causes rejects the neo-

Galtonian approach to scientific human psychology, but maintains the search for punctualprediction in M(Y ).

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In such game where the target is a law, any pre-scientific inspiring sourceis allowed for descriptive attempts, provided the result is a general fact(Ω, α, β)[c1, c2] with c1 and c2 as big as possible. Moreover, until no generalfact has been discovered, any substantive theory is, by definition, prematureas nomothetic theory. As Krause (2010) has stated,

If there is no SCCs [sufficient condition causes] in the subjectmatter domain of scientific human psychology, then there is noexplanatory science of human psychology, because then there can-not properly be said to be any definable psychological causes ofhuman psychological phenomena but, perhaps, only probabilisticpsychological predictors of such phenomena. (p. 59)

There is a philosophical argument for not expecting totally deterministicpsychological laws: if one admits that a totally deterministic law can becorroborated when observing psychological phenomena in the M(X)×M(Y )language, one accepts that there is no need for free will in some specificconditions (α). In other terms, such psychological phenomena obey whatSearle (1983) calls physical causality as opposed to intentional causality. Thecase of partial determinacy leaves room for free will as a defining feature ofpsychological systems, however they can be described.

Suppose a psychological law has been discovered, such that β is a sub-set of several values. Such law acknowledges the irreducible uncertainty of(human) behavior but bounds it within the fixed limits of β given α. Hence,such scientific knowledge implies that punctual prediction in M(Y ) is not ex-pected given the available scientific knowledge. If applied psychologists are tobe serving the social demand by proposing better expectations, they cannotargue that their better expectations are valid when applied to single cases.In the domain of psychological assessment, we contend that the P1-paradigmallows researchers interested in satisfying the social demand through psycho-logical measurements (or index-numerology according to Johnson, 1943) toestablish that quantitative psychological attributes can be measured becausemeasurement is possible without deterministic laws–even without partiallydeterministic laws (for a more detailed discussion, see Vautier et al., 2012).

As argued by Atlan (1986), the domination of the natural sciences rests ontheir technical efficiency. The technical efficiency in the social sciences relatesto decision making under uncertainty. It can be expected that a psychologicalscience programmed to demonstrate partial determinism of (human) behaviorshould provoke fierce resistance in the community of social scientists who

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justify its efforts with respect to social utility. But social utility should not bea sine qua non for advancing psychological research. If the scientific endeavordepends to an excessive extent on the social utility of its applications, itseems that the scientific endeavor has to be of the (neo-)Galtonian sort. Weadvocate room for disinterested research (see Michell, 1997), which entailsa publishing space for P2-researchers. We would enjoy being able to claimthat psychological laws of form (1), or equivalently, relations with graphsof form (17), have been discovered since the historical beginnings of thescientific endeavor in psychology, and that such claims can be qualified fortheir intrinsic scientific interest. Currently, to deserve publication, researcharticles have to be (i) socially interesting or (ii) motivated by rationalesformulated in a non-descriptive but substantive language to be perceivedas theoretically relevant. Taking into account the epistemological limits ofthese practices, this seems counter effective. To discover psychological laws(i.e., descriptive statements that are true for any individual person in givenconditions) we need to be able to develop a nomothetic research programme,and, for this, public knowledge of the existence of these ideas is of paramountimportance.

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